Removing untextured faces from UV mapping
You can't really remove the useless faces from the UV map but for example you can select them in the 3D view, Unwrap &gt; Reset, which will stack them, and put them out of the UV Editor grid:

Then select the rest of the mesh (Ctrli) and unwrap with the Smart UV Project option. Now the selected part occupy the whole grid, as expected (but it means that you'll need an image that will match the new unwrap):

If we select both the useful and useless part, as you see, the useless part keeps its location. You can put it over the second UV unwrap but it's more convenient to separate the 2 parts if you ever need to select one or the other. Note that the useless will inherit of the texture that will be underneath (an image texture is always infinitely repeated in the UV Editor) but you said that this part won't be seen so you don't have to care. If you want this part to be alpha 0% you need to put it over a part of your image that will be alpha 0%: